Lemma 7

Soit T un espace topologique localement connexe. Soit $\mathscr{U}$ un ensemble filtrant croissant de parties ouvertes de T, de réunion T. Pour $t\in T$ et $U\in \mathscr{U}$, notons $U_t$ la composante connexe de $t$ dans U si $t\in U$ et posons $U_t=\emptyset$ si $t\notin U$. Pour tout $t\in T,\bigcup_{U\in\mathscr{U}}U_t$ est la composante connexe de $t$ dans T.

For $t\in T$, put $C_t=\bigcup_{U\in\mathscr{U}}U_t$. We have $t\in C_t$. The sets $U_t$ are open and connected and the same is true of $C_t$ since we have $t\in U_t$ if $U_t=\not\emptyset$ (TG, I, p. 81, prop. 2). Let $u,v$ be points of T such that $C_u\cap C_v=\not\emptyset$. Let $t$ be a point of $C_u\cap C_v$; let $U\in \mathscr{U}$ be such that $t\in U_u$ and let $V\in \mathscr{U}$ be such that $v\in V_v$. Since $\mathscr{U}$ is increasing filtrant, there exists $W\in \mathscr{U}$ which contains $U\cup V$. Then, $W_u\cap W_v=\not\emptyset$, hence $W_u= W_v$. More generally, one has $W'_u= W'_v$ for every $W'\in \mathscr{U}$ such that $W\subset W'$, hence $C_u= C_v$. This proves that the sets of the form $C_t$ form a partition of T into open connected subsets. Consequently, for every $t\in T$, $C_t$ is the connected component of $t$ in T.